Limit point and limit circle trichotomy for Sturm-Liouville problems with complex potentials

Abstract

The limit point and limit circle classification of real Sturm-Liouville problems by H. Weyl more than 100 years ago was extended by A.R. Sims around 60 years ago to the case when the coefficients are complex. Here, the main result is a collection of various criteria which allow us to decide to which class of Sims' scheme a given Sturm-Liouville problem with complex coefficients belongs. This is subsequently applied to a second order differential equation defined on a ray in $\mathbb{C}$ which is motivated by the recent intensive research connected with $\mathcal{P}\mathcal{T}$-symmetric Hamiltonians.